Volume 72B, number 1 PHYSICS LETTERS 5 December 1977

H Y P E R F I N E I N T E R A C T I O N S I N N E G A T I V E P A R I T Y B A R Y O N S *

Nathan ISGUR Department of Physics, University of Toronto, Toronto, Canada

amd

Gabriel KARL Department of Physics, University of Guelph, Guelph, Canada

Received 11 October 1977

The hyperfine interaction between quarks suggested in chromodynamics can, without free parameters, explain the size and pattern of the splittings and the mixing angles observed experimentally in nonstrange, negative parity baryons.

There is growing support for the idea [ I ] that gluon- mediated quark-quark interactions inside hadrons are similar to photon-mediated interactions. De Rujula et al. [2] have shown very convincingly how the so- called Fermi contact term correlates the splitting of A-N, N-A and I~*-N in ground state baryons. More recently, Carlitz et al. [3] have shown that the same interaction accounts very well for the charge radius of the neutron* t . Apart from the Fermi contact term there is a second, so-called tensor term which is part of the hyperfine interaction. The neutron charge ra- dius and the ground-state mass differences are not sensitive in leading order to the tensor forces which mix L = 2 components into the L = 0 ground state. Contact and tensor forces operate side-by-side, how- ever, in negative pari ty baryons. We have found that the hyperfine interaction of chromodynamics can explain the pattern and size of the splittings and the mixing angles of negative parity nonstrange baryons.

The effective hyperfine interaction between two quarks 1 and 2 widely assumed in chromodynamics ( [2] , for a nice review, see [4]) simulates the magne- tic dipole-magnetic dipole interaction of electrodyna- mics:

* Research supported by the National Research Council of Canada.

, t Their considerations can be extended to the computation of the electric form factor of the neutron G~(q 2), which turns out to have the right size and to have one sign only (no nodes), in agreement with experiment.

/ / 1 2 = A {(8rr/3)S l "S 2 8 3 (p)

+ P-3(3S1" b S 2" b -S1"$2)} , (1)

where S 1 and S 2 are the spins of the two quarks, 21/2 p -= r 1 - r 2 is a vector joining them, and A is an overall constant depending on quark masses and the interaction strength ,2 . As is well known in atomic physics [see e.g. [5]), the two terms with relative strength as displayed in eq. (1) are two parts of a single physical interaction: the static interaction of two in- trinsic magnetic dipoles. The second term (often called the ' tensor ' par t ) averages to zero in orbital S states of the pair, and so is operative only when a pair has non-zero orbital angular momentum, while the first term (called the 'Fermi contact ' term) is operative only when the pair has zero orbital angular momentum. Our discussion is independent of A except when dealing with the size of splittings, and then we normalize A by the size o f the A-N mass difference.

The lowest mass negative pari ty baryons are assigned [6] in the quark model to a seventyplet of orbital angular momentum L = 1. Our discussion relies on this assignment. There are three pairs of quarks in a baryon, and in the simplest models*3, which we assume, the

,2 The interaction (1) corresponds to the single gluon exchange part of the potential. The needed size of A is roughly in ac- cord with a "color" coupling constant a s of order unity: A = as/(3 m2x/2), where m is the effective quark mass, for a pair of equal mass quarks separated by r12 = ~ .

,3 An example of this class of models is provided by three par- ticles connected by harmonic forces.

109

Volume 72B, number 1 PHYSICS LETTERS 5 Decembez 1977

~800 -

t / 0 0 -

i

,~oo- ~

q3

1500l

N 'I12- N" 5/2 N ' 5 / 2 & ' l l ~ - A ' 3 / 2 -

Fig. I. Comparison between theory and experiment for non- strange P-wave baryons. The unperturbed mass was chosen to be 1610 MeV. The shaded regions indicate the interval of masses quoted in the Baryon Table of the Particle Data Group, with the exception of the two star 3- resonance at around 1740, which we have estimated.

orbital angular momentum L = 1 resides in a single pair while the other two pairs have zero orbital angular momenta. Thus in one pair the tensor term is operative, while in the other two the contact force represents the hyperfine interaction. The negative parity baryons are as a result a good place to test both pieces of the hyperfine interaction.

The nonstrange components of the (70, 1 -} super- multiplet consist of five isodoublets - two of quark spin S = 1/2 and three of quark spin S = 3/2 - and two isoquartets of S = 1/2. We first discuss the splitting of these states induced by the contact term. To calcu- late a matrix element we must consider the interaction in all three pairs; by the overall symmetry of the wave- functions we can instead mult iply a specific pair by a factor of three (see e.g. [7}). We find:

(N;3 . .3 ; j ) = z r A ( ~ M i 5 3 ( p ) l ~ M ) ' 1, JIHcontact I N, ~ 1

, . . , --TrA(~MI53(p)[~j~M), (N; ~- 1, J[HcontactlN, y 1 ; J ) =

(A; ½ 1; JIHcontactlA; ~- 1; J) = ~rA(@)MI63 (P)[ ~)M), (2)

where ~;~M is the component of the spatial wavefunc- tion which is even under the transposition of particles 1 and 2 and has L = 1, L z = M. The odd component C o does not contribute to the contact term since it has to vanish when = 0. The expectation value is indepen- dent of M. The splitting pattern induced by this term is in qualitative agreement with experiment [2] : the

states are split into two groups: a pair of N*'s with JP = 1 / 2 - , 3 / 2 - and a second, higher set of states consisting of three N*'s with JP = 1 / 2 - , 3 / 2 - , 5 / 2 -

and two A*'s o f J P = 1 / 2 - , 3 / 2 - . Moreover, the size

of the splitting

M(3/2) - M(1/2) : 2 A~r(ff~MIf3(p)I~M), (3)

can be correlated with the size of the splitting between the A (1236) and the nucleon N (940). If we take har- monic oscillator wavefunctions (see e.g. [8]) we find:

A-N = 4n'A(ffS0163(p)l~S0 ) ~ ~ = 4Ao~3~'-1/2 (4) and ( ~ 1 1 8 3 ( P ) I ~ I > = a3~'-3/2 ' (5)

where c~ is a harmonic oscillator parameter. From eqs. (3), (4) and (5):

' (A-N) -~ 150 MeV (6) M(3/2) - M(1/2) = g

in agreement with the experimental splitting, as eva- luated for example from M [N*(1670)] - M [N*(1520)] -~ 1 6 7 0 - 1520 = 150 MeV. The main incorrect predic- tion, were we to stop at this point , is the expectat ion (see eq. (2)) that both low lying states N*(1520) and N*(1535) are pure quark spin states S = 1/2, in disagree- ment with experiment [9].

We now include the tensor force in our discussion as we must. The tensor force has S = 2, so that its ma- trix elements vanish between any two S = 1/2 states. However, the other matrix elements are nonzero. The direct computat ion of the matr ix elements of the ten- sor force is somewhat lengthy; we found the following identity [10] useful in reducing the labour involved:

( 1 S J I p - 3 ( 3 S1 " PS 2 " D - S 1 "$2)11S 'J}

= ( - - ) J -1 -S ' ( 3 (2S+ 1)) 1/2 W(11SS'; 2 J )

X (1]}½x/'3p-3~+D+II1)(SII½,v/3S1-S2-IIS'), (7)

where W is a Racah coefficient, and the last two factors are reduced matrix elements of the tensors whose +2 and - 2 components respectively are displayed. Using this identi ty and standard 70-plet wavefunctions we find for the total contribution of tensor forces in all three pairs:

3 5 3 5 (N; 1 ~ ~-IHtensorlN; 1 ~- -f) = 3 a / 4 , (8 a)

3 3 3 _ (N; 1 ~ ~ IHtensor[N; 1 ~" ~) - - 3 a , (Sb)

110

Volume 72B, number 1 PHYSICS LETTERS 5 December 1977

(N;1}}[HtensorlN; 3 1 1 7 : ) = 15a/4, (8c)

(N;1 }3~.lHtensor IN ,. 1 7' }) = -(3/4)(5/2) 1/2a , (8d)

3 1 I l (N; 1 -ff 7 [Htensor[ N; 1 ~" 7) = 15a/4 , (8e)

with (9)

a -= ½ A ( ~ I Ip-3 (3 cos20o - I ) 1 ~ 1 )= -(4/15)Aot3n -l/2,

where we have assumed that only one pair has nonzero orbital angular momentum, and evaluated the 'radial' matrix element with harmonic oscillator wavefunc- tions as in eq. (5). Note that while the matrix elements of the contact term (2) depend only on the even (X) component of ~, the matrix elements of the tensor term depend on the odd (P) component. From eqs. (2), (5), (8) and (9) we obtain the full effect of the hyperfine interaction H in the nonstrange states, written below in units (Ac~3~r - I / 2 ) = 4 -1 (M a - MN) -~ 75 MeV.

In the zS* sector we have from eq. (2) alone

1 3 ~ A * I 3~_ ( A ' 7 7 **hyp ~ 7 T , - 1 --~ 75 MeV

* 1 1 . 1 1 _ (A 771Httyp}A 7 7 ) - 1 ~ 7 5 M e V ,

in the J = 5/2 N* sector from eqs. (2), (5), (8a) and (9)

N*3 5 yp} N * & 5 ' = 1 - 1 4 ~ 7~-}Hh ~" 2 2- ?-=-~ 60MeV (10)

in the J = 3/2 N* sector from eqs. (2), (5), (8b, d) and (9) the interaction is

, , , . .~3) i~, 77~ (ll) H h y p \ M * t 3 ,1 3 ,, ~'7 \ l / x / ~ - 1 / \ N ~ - ~ ] '

while in the J = 1/2 N* sector from eqs. (2), (5), (8c,e) and (9) it is

H , - - n y p \ . i l I 1 \ M * 1 3 1 " N 7 7 - -, ~ 7 -

It remains to diagonalize the J = 3/2 and 1/2 N* sectors to find the physical eigenstates. The eigenvalues in the J = 3/2 sector are:

~3/2 = 10_1(4+ ~ ) ~ 1.835 -~ +135 MeV,

- - 1 . 0 3 5 - ~ - 7 5 MeV, (13)

with the eigenstate corresponding to the lower eigen- value X (identifed with N*(1520)) being:

IN*(1520)) = - s i n 0dl-~}) + cos 0d [~ r ) 3 3 1 3 = - 0 . 1 1 1 ~ 7 ) + 0 .99417~) , (14)

corresponding to

0d= arctan (X,~/(14 + 2 x / ~ ) ) --~ +6.3 ° ,

to be compared with the empirical mixing, found by Hey, Lichtfield and Castunore from analysing decay data [9] *4

IN*(1520)> -~ -0.181}}) +0.981 ' 3 77>, (15)

corresponding to 0 a -~ +10 °. The eigenvalues in the J = 1/2 sector are:

1/2 _ 2_1 (_ 1 -+ x/5-) -~ + 0.618 ~ 45 MeV k+ --

(16) --1.618 - ~ - 1 2 0 MeV,

with the state of low mass [identified with the N*(1530)] being

IN*(1530)) = - s in 0sl ~ 1 l 7 7 )+ cos 0s177)

3 1 1 1 = 0.526177) + 0.85117-¢), (17)

corresponding to a mixing angle of

0 s = -arc tan½ ( x / 5 - 1) ~- - 3 1 . 7 ° ,

to be compared with the empirical mixing, found very accurately by Hey, Lichtfield and Cashmore to be [9]:

3 1 IN*(1530))= 0.5317~-) + 0.851½½), 0 - ~ - 3 2 °. (18)

The agreement of the composition of states (14) and (17), predicted by the Hamiltonian (1), with the experimental determinations (15) and (18) respectively is very striking. The predictions (14) and (17) are inde- pendent of any choice of parameters such as the coupl- ing strength A or choice of harmonic oscillator con- stant a. These mixings are brought about by the pre- sence of tensor terms and their magnitude depends only on the relative size of contact and tensor terms which is given (apparently correctly!)in eq. (1).

The pattern and size of the splittings predicte d by eqs. (10), (13) and (16) with the constant (Ac~37r -1/2) determined as in eq. (4) is also in agreement with expe- riment as illustrated in fig. 1, where the broad bands represent the limits within which the Particle Data Group feel the masses of the various resonances are likely to lie. The degree of agreement is surprisingly ...J

,4 Our conventions as to the relative sign between S = 1/2 and S = 3/2states are the same as those of Hey et al. [9]. There- fore, the comparison of relative signs is meaningful.

111

Volume 72B, number 1 PHYSICS LETTERS 5 December 1977

good considering that the only parameter at our dis- posal, Aa3cr - 1/2, was determined from a splitting in

an entirely different supermultiplet . Previous group-theoretical analyses of the spectrum

of negative parity baryons [ 11 ] have ignored tensor terms but kept instead spin-orbit terms * s. These ana-

lyses had no difficulty in fitting the mass spectrum (by appropriate choice of parameters) bu t failed to obta in any sizeable mixing as observed [9] in the

J = I /2 sector. At the risk of annoying the reader, we repeat that from the point of view of chromodynamics it is on the contrary mandatory that the tensor and contact terms be treated together, as they have the same physical origin. Nevertheless in analogy with electrodynamics one generally expects spin-orbit coupling to be present as well .6 . Since we neglected spin-orbit coupling completely and obtained satisfac- tory agreement with exper iment , it is relevant to ask how strong a spin-orbit coupling one could tolerate without destroying this agreement. To normalize our answer we take as s tandard the Breit Hamil tonian [2, 13]. For a 1/r potent ial be tween equal mass quarks

this predicts a coupling (3 A/2 O 3)L 12" ($1 + $2) with the same A as in eq. (1). If spin-orbit coupling were present with this strength, it would shift the N * 5 / 2 - state upward by ~ 2 2 5 MeV relative to the unshif ted A* states .7 . Since in reality the J = 5/2 N* and the A*'s are nearly degenerate the spin-orbit coupling if present at all can be no stronger than about 10% of the value predicted by a Breit Hamil tonian [12] *8 This conclusion is very similar to one reached by Schnitzer [13} in this analysis of the P wave states of charmonium*9 '~° .

,5 From a phenomenological point of view there is of course no reason at all to correlate the tensor and contact forces to each other. This is the main advantage of a dynamical prescription such as that suggested by chromodynamics.

,6 Recall, however, that while the hyperfine interaction is the interaction of two intrinsic colour magnets, the spin-orbit coupling arises from a distinct physical mechanism.

,7 An analysis including full-strength spin-orbit coupling (see [ 14 ]) clearly demonstrates that this option is untenable.

,8 Note that the contribution of the tensor term is not small when compared to the contact term, as claimed erroneously in [12]. For example in the S = 3•2, J = 1/2 state the tensor term exactly cancels the contribution of the contact term - leading to the zero total matrix element.

,9 Similar ideas about P wave states in charmonium were also discussed by K. Johnson in a lecture in December 1976.

In looking critically at our results it is nex t relevant to ask whether they are sensitive to the use of the harmonic oscillator model. We stress that our conclu- sions do no t depend on the use of harmonic oscillator forces. The model was employed only to generate wavefunct ions to compute matr ix elements, and as shown by Gromes and Stamatescu [ 12] harmonic oscillator wavefunct ions are a good approximat ion to the e igenfunct ions of low-lying states of a system bound by Coulomb-plus-l inear potentials .

We conclude that the QCD inspired hyperf ine inter- action (1) is dominan t in determining the splittings and mixing angles in P-wave baryons. I t remains to be seen whether such effects can be rigorously derived in Quan tum Chromodynamics .

We thank F.E. Close and M. Kugler for conversa- tions which helped us into this line of thought , and the

Directors of the Les Houches Summer School (1976) for their splendid physical facilities which helped ini- tiate our work.

References

[1} M.Y. Han and Y. Nambu, Phys. Rev. 139B (1965) 1006; H. Fritzsch, M. GeU-Mann and H. Leutwyler, Phys. Lett. 47B (1973) 365.

[2] A. De Ru]ula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147; see also: T. De Grand, R.L. Jaffe, K. Johnson and I. Kiskis, Phys. Rev. D12 (1975) 2060.

[3] R.D. Carlitz, S.D. Ellis and R. Savit, Phys. Lett. 68B (1976) 443; see also: N. Isgur, Lectures at the XVIIth Cracow School of Theoretical Physics, to be published in Acta Phys. Polon.

[4] .I.D, Jackson, Lectures on the new particles, Proc. 1976 Summer Institute on Particle Physics, Stanford Linear Accelerator Center, Report No. 198 (1976); see also: J. Pumplin, W. Repko and A. Sato, Phys. Rev. Lett. 35 (1975) 1538; H.J. Schnitzer, Phys. Rev. Lett. 35 (1975) 1540; Phys.

, lo The splitting between the S wave states qJ and ~c is deter- mined by the contact term in mesons. It is amusing to note note that if one takes the "experimentally determined" q~ wavefunctions at zero relative separation (which rise roughly in proportion to the reduced mass of the q~ pair), then the large q~-~c mass difference can be understood in terms of the o-~r spfitting even though the contact term is inversely proportional to the product of the quark masses.

112

Volume 72B, number 1 PHYSICS LETTERS 5 December 1977

Rev. D13 (1976) 74; R, Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Lett. 57B (1975) 445.

[5] H.A. Bethe and E.E. Salpeter, in: Handbuch der Physik, vol. X X X V (Springer, Berlin, 1957) p. 267.

[6} R.H. Dalitz, Proc. Summer School Les Houches (1965). [7] L.A. Copley, G. Karl and E. Obryk, Nucl. Phys. B13

(1969) 303. [8] G. Karl and E. Obryk, Nucl. Phys. B8 (1968) 609. [9] A.J.G. Hey, P.J. Litchfield and R.J. Cashmore, Nucl.

Phys. B95 (1975) 516; see also: D. Faiman and D.E. Plane, Nucl. Phys. B50 (1972) 379.

[10] D,M. Brink and G.R. Satchler, Angular momentum (Oxford University Press, Oxford, 1962).

[11] D.R. Divgi and O.W. Greenberg, Phys. Rev. 175 (1968) 2024; D. Horgan, Nucl. Phys. B71 (1974) 514; For a recent review see: R.R. Horgan, in: Proc. Topical Conf. on Baryon Resonances, eds. R.T. Ross and D.H, Saxon, Rutherford Laboratory S.R,C. Chilton, Didcot, UK (1976) p. 434.

[12[ D. Gromes and I.O. Stamatescu, Nucl. Phys. B112 (1976) 213.

[131 H.J. Schnitzer, Phys. Lett. 65B (1976) 239. [14] W. Celmaster, Phys. Rev. D15 (1977) 1391.

113